A071752 Expansion of (1+x^4*C^3)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 3, 9, 28, 91, 303, 1028, 3542, 12363, 43628, 155414, 558144, 2018750, 7347281, 26888610, 98888730, 365292555, 1354757040, 5042499990, 18830130360, 70527871530, 264886906830, 997372693824, 3764155637772, 14236915323806, 53955477752168, 204864378666108

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = 3*binomial(2*n,n)*(17*n^3+14*n^2-13*n+14)/(2*(n+1)*(n+2)*(n+3)*(2*n-1)) for n > 1. - Tani Akinari, Aug 06 2025

a(n) ~ 51 * 4^(n-1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 04 2025

Mathematica

c = (1 - (1 - 4*x)^(1/2))/(2*x); (1 + x^4*c^3)*c^3 + O[x]^25 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 27 2017 *)
a[n_] := 3 * Binomial[2*n, n] * (17*n^3 + 14*n^2 - 13*n + 14)/(2*(n + 1)*(n + 2)*(n + 3)*(2*n - 1)); a[0] = 1; a[1] = 3; Array[a, 30, 0] (* Amiram Eldar, Oct 04 2025 *)

Prog

(Maxima) a(n):=if n<2 then 3^n else 3*binomial(2*n, n)*(17*n^3+14*n^2-13*n+14)/(2*(n+1)*(n+2)*(n+3)*(2*n-1));
makelist(a(n), n, 0, 50); /* Tani Akinari, Aug 06 2025 */

Crossrefs

Cf. A000108.

Sequence in context: A047047 A071744 A071748 * A071756 A176673 A215007

Adjacent sequences: A071749 A071750 A071751 * A071753 A071754 A071755

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved